The present invention relates to a new and distinct cultivar of chrysanthemum plant, botanically known as Chrysanthemum×morifolium Ramat., commercially known as a garden mum, and hereinafter referred to by the cultivar denomination ‘Zanmufrenzy’. ‘Zanmufrenzy’ is a product of a breeding and selection program for outdoor pot mums (garden mums) which had the objective of creating new cultivars with a double type inflorescence, a natural season flowering date around September 3 (week 36), blooming for a period of 5 weeks. ‘Zanmufrenzy’ is a seedling resulting from a cross of the female parent id 24446 with the male parent id 27629. Plants of the new cultivar ‘Zanmufrenzy’ differ from plants of the female parent in the color of the ray-florets; the ray-florets of the seedling are purple, while those of the female parent are red. Plants of the new cultivar ‘Zanmufrenzy’ differ from plants of the male parent in the following characteristics. (1) Color ray-florets. And (2) Inflorescence type. (1) The ray-florets of the seedling are less intensively purple colored than those of the male parent. (2) The inflorescence type of the seedling is double, while that of the male parent is semi-double. The new and distinct cultivar was discovered and selected as a flowering plant by Wilhelmus Bernardus Blom on a cultivated field in Rijsenhout, The Netherlands in 2008. The first act of asexual production of ‘Zanmufrenzy’ was accomplished when vegetative cuttings from the initial selection in 2008 were propagated further in a controlled environment in Rijsenhout, The Netherlands. The new cultivar has been found to retain its distinctive characteristics through successive propagations.